Chapter 6.3, Problem 84E

To determine

(a)To calculate; the value of $P(Y\ge 10)$

Answer to Problem 84E

  $\text{P(Y }\ge \text{ 10) = P(X }\le \text{ 2) = }55.83\%$

Explanation of Solution

Given;

  $\begin{array}{l}n=12\\ p=0.20\end{array}$

Explanation;

  $\begin{array}{l}\text{Definition Binomial Probability:}\\ $ \text{P(X = k) = }\left(\begin{array}{l}n\\ k\end{array}\right)\times p^k\times {(1-p)}^{n-k}$\end{array}$

In the given case, if the number of people which are identified by lie detector as speaking truth is 10 or more, then the number of people being deceptive is 2 or less than 2. The reason being, as there are 12 number of people in total.

  $\text{P(Y }\ge \text{ 10) = P(X }\le \text{ 2) = P(X = 0) + P(X = 1) + P(X = 2)}\approx 0.5583\approx 55.83\%$

To determine

(b)To calculate; the value of ${\mu }_Y$ and ${\sigma }_Y$

Answer to Problem 84E

  ${\mu }_Y=9.6$

  ${\sigma }_Y=1.3856$

Explanation of Solution

Given;

  $\begin{array}{l}n=12\\ p=0.20\end{array}$

Explanation;

It can be noted that Y is similar to X, where the only difference is that the values of success and failure are interchanged.

  $\begin{array}{l}{p}_Y=1-p_X=1-0.20=0.80\\ n=12\end{array}$

  ${\mu }_Y=n\times p=12(0.80)=9.6$

  ${\sigma }_Y=\sqrt{(n\times p)(1-p)}=\sqrt{12(0.80)(1-0.80)}=1.3856$